3.62.46 \(\int \frac {5+(-3+2 x^2) \log ^2(\log (3))}{x^2 \log ^2(\log (3))} \, dx\)

Optimal. Leaf size=23 \[ \frac {3}{x}+2 x-\frac {3+\frac {5}{x}}{\log ^2(\log (3))} \]

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 0.74, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 14} \begin {gather*} 2 x+\frac {3-\frac {5}{\log ^2(\log (3))}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + (-3 + 2*x^2)*Log[Log[3]]^2)/(x^2*Log[Log[3]]^2),x]

[Out]

2*x + (3 - 5/Log[Log[3]]^2)/x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {5+\left (-3+2 x^2\right ) \log ^2(\log (3))}{x^2} \, dx}{\log ^2(\log (3))}\\ &=\frac {\int \left (2 \log ^2(\log (3))+\frac {5-3 \log ^2(\log (3))}{x^2}\right ) \, dx}{\log ^2(\log (3))}\\ &=2 x+\frac {3-\frac {5}{\log ^2(\log (3))}}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.83 \begin {gather*} \frac {3}{x}+2 x-\frac {5}{x \log ^2(\log (3))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + (-3 + 2*x^2)*Log[Log[3]]^2)/(x^2*Log[Log[3]]^2),x]

[Out]

3/x + 2*x - 5/(x*Log[Log[3]]^2)

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fricas [A]  time = 1.08, size = 24, normalized size = 1.04 \begin {gather*} \frac {{\left (2 \, x^{2} + 3\right )} \log \left (\log \relax (3)\right )^{2} - 5}{x \log \left (\log \relax (3)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-3)*log(log(3))^2+5)/x^2/log(log(3))^2,x, algorithm="fricas")

[Out]

((2*x^2 + 3)*log(log(3))^2 - 5)/(x*log(log(3))^2)

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giac [A]  time = 0.14, size = 28, normalized size = 1.22 \begin {gather*} \frac {2 \, x \log \left (\log \relax (3)\right )^{2} + \frac {3 \, \log \left (\log \relax (3)\right )^{2} - 5}{x}}{\log \left (\log \relax (3)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-3)*log(log(3))^2+5)/x^2/log(log(3))^2,x, algorithm="giac")

[Out]

(2*x*log(log(3))^2 + (3*log(log(3))^2 - 5)/x)/log(log(3))^2

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maple [A]  time = 0.07, size = 20, normalized size = 0.87

method	result	size
risch	\(2 x +\frac {3}{x}-\frac {5}{\ln \left (\ln \relax (3)\right )^{2} x}\)	\(20\)
gosper	\(\frac {2 \ln \left (\ln \relax (3)\right )^{2} x^{2}-5+3 \ln \left (\ln \relax (3)\right )^{2}}{\ln \left (\ln \relax (3)\right )^{2} x}\)	\(29\)
default	\(\frac {2 x \ln \left (\ln \relax (3)\right )^{2}-\frac {-3 \ln \left (\ln \relax (3)\right )^{2}+5}{x}}{\ln \left (\ln \relax (3)\right )^{2}}\)	\(30\)
norman	\(\frac {\frac {3 \ln \left (\ln \relax (3)\right )^{2}-5}{\ln \left (\ln \relax (3)\right )}+2 x^{2} \ln \left (\ln \relax (3)\right )}{\ln \left (\ln \relax (3)\right ) x}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-3)*ln(ln(3))^2+5)/x^2/ln(ln(3))^2,x,method=_RETURNVERBOSE)

[Out]

2*x+3/x-5/ln(ln(3))^2/x

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maxima [A]  time = 0.35, size = 28, normalized size = 1.22 \begin {gather*} \frac {2 \, x \log \left (\log \relax (3)\right )^{2} + \frac {3 \, \log \left (\log \relax (3)\right )^{2} - 5}{x}}{\log \left (\log \relax (3)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-3)*log(log(3))^2+5)/x^2/log(log(3))^2,x, algorithm="maxima")

[Out]

(2*x*log(log(3))^2 + (3*log(log(3))^2 - 5)/x)/log(log(3))^2

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mupad [B]  time = 0.06, size = 22, normalized size = 0.96 \begin {gather*} 2\,x+\frac {3\,{\ln \left (\ln \relax (3)\right )}^2-5}{x\,{\ln \left (\ln \relax (3)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(3))^2*(2*x^2 - 3) + 5)/(x^2*log(log(3))^2),x)

[Out]

2*x + (3*log(log(3))^2 - 5)/(x*log(log(3))^2)

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sympy [A]  time = 0.09, size = 27, normalized size = 1.17 \begin {gather*} \frac {2 x \log {\left (\log {\relax (3 )} \right )}^{2} + \frac {-5 + 3 \log {\left (\log {\relax (3 )} \right )}^{2}}{x}}{\log {\left (\log {\relax (3 )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-3)*ln(ln(3))**2+5)/x**2/ln(ln(3))**2,x)

[Out]

(2*x*log(log(3))**2 + (-5 + 3*log(log(3))**2)/x)/log(log(3))**2

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